Table Of ContentAtlantis Transactions in Geometry
Stefan Haesen
Leopold Verstraelen
Editors
Topics in Modern
Differential
Geometry
Atlantis Transactions in Geometry
Volume 1
Series editor
Johan Gielis, Antwerpen, Belgium
The series aims at publishing contemporary results in geometry including large
partsofanalysisandtopology.Theserieswillpublishbooksofboththeoreticaland
appliednature. Theoreticalvolumeswill focus among other topicson submanifold
theory, Riemannian and pseudo-Riemannian geometry, minimal surfaces and
submanifolds in Euclidean geometry. Applications are found in biology, physics,
engineering and other areas.
More information about this series at http://www.atlantis-press.com/series/15429
Stefan Haesen Leopold Verstraelen
(cid:129)
Editors
Topics in Modern
Differential Geometry
Editors
StefanHaesen Leopold Verstraelen
Department ofTeacher Education Department ofMathematics
ThomasMore University College University of Leuven
Vorselaar Leuven
Belgium Belgium
Atlantis Transactionsin Geometry
ISBN978-94-6239-239-7 ISBN978-94-6239-240-3 (eBook)
DOI 10.2991/978-94-6239-240-3
LibraryofCongressControlNumber:2016955687
©AtlantisPressandtheauthor(s)2017
This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by
anymeans, electronic or mechanical, including photocopying, recording or any information storage
andretrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher.
Printedonacid-freepaper
Preface
In 2008 and 2009, the Simon Stevin Institute for Geometry participated in the
organization of Ph.D. courses at the universities of Leuven (Belgium), Kragujevac
(Serbia), Murcia (Spain) and Brasov (Romania). Besides the main course lectures
on “Natural geometrical intrinsic and extrinsic symmetries”, there were invited
shortlecturesonavariedselectionoftopicsindifferentialgeometry.Severalofthe
lecturers were able to find the time to prepare their talks for the publication in this
book.
Oursincerethanksgotoallthestudentswhoparticipatedatthevariouscourses,
toProf.Dr. R.Deszcz andDr.A.Albujerwho taughtseveralofthemainlectures,
toProf. Dr. L. Alías, Prof. Dr. F. Dillen,Dr. J. Gielis, Prof. Dr. I.Mihai,Prof. Dr.
M. Petrović-Torgašev and Prof. Dr. E. Stoica for the local organization of the
courses and to all the invited lecturers.
Vorselaar, Belgium Stefan Haesen
Leuven, Belgium Leopold Verstraelen
February 2010
v
Contents
The Riemannian and Lorentzian Splitting Theorems... .... ..... .... 1
José Luis Flores
Periodic Trajectories of Dynamical Systems Having
a One-Parameter Group of Symmetries . .... .... .... .... ..... .... 21
Roberto Giambò and Paolo Piccione
Geometry and Materials. .... ..... .... .... .... .... .... ..... .... 37
Bennett Palmer
On Deciding Whether a Submanifold Is Parabolic or Hyperbolic
Using Its Mean Curvature ... ..... .... .... .... .... .... ..... .... 49
Vicente Palmer
Contact Forms in Geometry and Topology... .... .... .... ..... .... 79
Gheorghe Pitiş
Farkas and János Bolyai .... ..... .... .... .... .... .... ..... .... 95
Mileva Prvanović
Spectrum Estimates and Applications to Geometry .... .... ..... .... 111
G. Pacelli Bessa, L. Jorge, L. Mari and J. Fábio Montenegro
Some Variational Problems on Curves and Applications.... ..... .... 199
Angel Ferrández
Special Submanifolds in Hermitian Manifolds .... .... .... ..... .... 223
Ion Mihai
An Introduction to Certain Topics on Lorentzian Geometry. ..... .... 259
Alfonso Romero
vii
The Riemannian and Lorentzian Splitting
Theorems
JoséLuisFlores
Abstract In these notes we are going to briefly review some of the main ideas
involved in the formulation and proof of the Riemannian and Lorentzian Splitting
Theorems.Wewilltrytoemphasizethesimilaritiesanddifferencesappearedwhen
passing from the Riemannian to the Lorentzian case, and the way in which these
difficultiesareovercomebytheauthors.
1 Introduction
ThesplittingprobleminRiemannianandLorentzianGeometryiscloselyrelatedto
the idea of “rigidity” in Geometry. So, in order to introduce this problem, first we
aregoingtodedicatesomelinestorecallthisimportantnotion.
Assume that we are interested in studying some Riemannian manifold (M,g).
Usually,itisveryusefultocompareitwithsomemodelspaceM ,i.e.acomplete1-
K
connectedRiemannianmanifoldofconstantsectionalcurvatureK.Infact,therearea
seriesofresultswhichensurethat(M,g)willretainglobalgeometricalpropertiesof
M undercertainstrictcurvatureboundsfor(M,g)intermsofK.Evenmore,under
K
theseconditions,itisusuallypossibletoconcludethatMwillalsoretaintopological
propertiesofM .Anaturalquestionwhicharisesfromthissituationis,whathappen
K
when one relaxes the condition of “strict” curvature inequality to some “weak”
curvatureinequality?Itisnotdifficulttorealizethat,underthesenewhypotheses,
the conclusion may not hold any more. This is clearly illustrated by the following
simpleobservation:thereisacrucialdifferencebetweenthetopologyofthesphere
TheauthorisgratefultotheorganizersoftheInternationalResearchSchoolonDifferential
GeometryandSymmetry,celebratedatUniversityofMurciafromMarch9to18,2009,for
givinghimtheopportunitytodeliveralectureonwhichthesenotesarebased.Partiallysupported
bySpanishMEC-FEDERGrantMTM2007-60731andRegionalJ.AndalucíaGrantP06-FQM-
01951.
B
J.L.Flores( )
DepartamentodeÁlgebra,GeometríayTopologíaFacultaddeCiencias,
UniversidaddeMálagaCampusTeatinoss/n,29071Málaga,Spain
e-mail:fl[email protected]
©AtlantisPressandtheauthor(s)2017 1
S.HaesenandL.Verstraelen(eds.),TopicsinModernDifferentialGeometry,
AtlantisTransactionsinGeometry1,DOI10.2991/978-94-6239-240-3_1
2 J.L.Flores
(K >0) and that of the Euclidean space (K ≡0), even for spheres of radius very
big,andso,withcurvatureveryclosetothenullcurvatureoftheEuclideanspace.
However,arelevantpropertystillholds:aconclusionwhichbecomesfalsewhenone
relaxesthe“strict”curvatureconditiontoa“weak”curvatureconditionusuallycan
beshowntofailonlyunderveryspecialcircumstances!Thisimportantidea,usually
referredas“rigidity”inGeometry,isroughlysummarizedinthefollowingprototype
result:
PrototypeRigidityTheorem:IfM satisfiesa“weak”curvaturecondition,andthe
geometric restriction derived from the corresponding “strict” curvature condition
doesnotholdanymore,then M mustbe“veryspecial”.
Inorder torelatethesplittingproblemtothisprototyperigiditytheorem,letus
recallthefollowingresultbyGromollandMeyer[17]:
Theorem1.1 (Gromoll,Meyer)AcompleteRiemannianmanifold(M,g)ofdimen-
sionn ≥2suchthat Ric(v,v)>0forallv ∈TM isconnectedatinfinity.
This is a typical result where a strict curvature inequality (Ric(v,v)>0) implies
atopologicalrestrictiononthemanifold(connectednessatinfinity).Now,suppose
thatwereplacethestrictcurvaturecondition Ric(v,v)>0bytheweakcurvature
condition Ric(v,v)≥0and,consequently,weassumethat(M,g)failstobecon-
nectedatinfinity.Since M iscomplete,nowonecanensuretheexistenceofaline
joining any two different ends of M. Under these new hypotheses, Cheeger and
Gromoll proved that (M,g) must be isometric to a product manifold. So, this is a
typicalrigiditytheoreminthesensedescribedabove.ThisresultanditsLorentzian
versionconstitutethecentralsubjectofthesenotes.
In the next section we will establish with precision the Riemannian Splitting
Theorem. We will also provide some brief comments about the initial motivation
andtheprecedentsofthetheorem.Finally,wewillintroducesomebasicnotionsand
resultswhichwillbeusedlaterintheproof.InSect.3wewilloutlinethemainideas
involvedintheoriginalproofbyCheegerandGromoll.Thisproofstronglyusesthe
theory of elliptic operators, and, indeed, it is stronger than actually needed. So, in
Sect.4wewilldescribeanalternativeproofofthesameresult,givenbyEschenburg
andHeintze,whichminimizestheuseoftheelliptictheory.Thissecondapproach
willberelevantforusbecauseitintroducesanewviewpointusefulfortheproofof
theLorentzianversionofthetheorem.InSect.5,wewillrecallsomebasicnotions
andresultsfromLorentzianGeometry.Afterthat,wewillestablishtheLorentzian
Splitting Theorem in Sect.6, providing some brief comments about the main hits
inthehistoryofitssolution.TheproofofthisresultwillbestudiedinSect.7.We
willessentiallyfollowtheargumentsgivenbyGallowayin[14]:aftersomeprevious
technical lemmas in Subsects.7.1–7.3, the proof will be delivered in six steps in
Subsect.7.4.Finally,inSect.8wewillrecallarelatedopenproblemwithaphysical
significance,theBartnik’sConjecture.
TheRiemannianandLorentzianSplittingTheorems 3
2 RiemannianSplittingTheorem
TheRiemannianSplittingTheoremcanbestatedinthefollowingway[7]:
Riemannian Splitting Theorem (Cheeger, Gromoll) Suppose that the
Riemannianmanifold(M,g),ofdimensionn ≥2,satisfiesthefollowingconditions:
(1) (M,g)isgeodesicallycomplete,
(2) Ric(v,v)≥0forallv ∈TM,
(3) M has a line (i.e. a complete unitary geodesic γ :R→(M,g) realizing the
distancebetweenanytwoofitspoints).
Then M is isometric to the product (M,g)∼=(Rk ×M ,g ⊕g ), k >0, where
1 0 1
(M ,g )containsnolinesandg isthestandardmetriconRk.
1 1 0
This is a very important result which has been extensively used in Riemannian
Geometry in the last decades. An important precedent of this result is due to
Topogonov [23], who obtained the same thesis under the more restrictive curva-
tureassumption of nonnegative sectional curvature. The proof of theTopogonov’s
result lies on the Triangle Comparison Theorem by the same author. The original
motivationfortheCheegerandGromoll’sresultwasthenecessitytoextendtheexist-
ingresultsconcerningthefundamentalgroupofmanifoldsofnonnegativesectional
curvature[8]tothecaseofnonnegativeRiccicurvature.Inparticular,theyneeded
asplittingtheoremundertheweakerhypothesisofnonnegativeRiccicurvature,for
whichtheTopogonov’sTriangleComparisonTheoremdoesnotwork.Afirstsplit-
tingresultofthistypewereobtainedbyCohn-Vossenin[9].However,thegeneral
resultrequiredtotallynewarguments,whichwerenotdevelopedtillthepublication
oftheremarkablepaper[7].
InordertodescribetheproofoftheCheeger–GromollSplittingTheorem,firstly
weneedtointroducesomepreviousnotions,whichareofinterestbyitself:
Byarayγwewillunderstandanunitarygeodesicdefinedon[0,∞)whichrealizes
thedistancebetweenanyofitspoints.Then,theBusemannfunction(associatedto
γ)isdefinedasthefunctionb : M →R3obtainedfromthelimit
γ
b (·):= lim(r −d(·,γ(r))), (1)
γ
r→∞
whered isthedistanceassociatedtotheRiemannianmetricg.Itisnotdifficultto
provethatpreviouslimitalwaysexists(isfinite)andtheresultingfunctioniscontin-
uous.Infact,thelimit(1)existsandisfinitebecause,fromthetriangleinequality,
themap
r (cid:9)→b (p)=r −d(p,γ(r))
r
isnondecreasing
r −r =d(γ(r ),γ(r ))≥d(p,γ(r ))−d(p,γ(r )) if r ≤r
2 1 1 2 2 1 1 2
andboundedabove
